Math Analysis Problem Set 18-01 Use what you’ve learned in class to answer these questions. You’ve definitely learned enough to do them…you just might need to think about it a lot… 1. Given the points A = ( 5,−2,6 ) , B = ( −3,5, 4 ) , and C = ( 4,−1,8 ) , D = ( −4, 3,2 ) , and E = ( 6,−4,6 ) . a. Write the equation of a plane perpendicular to the line through A and B containing the point A. b. Write the equation of a plane perpendicular to the line through A and B containing the point B. c. Find the distance between the two planes. (Remember, distance is measured perpendicularly!) d. Find a vector orthogonal to the plane containing the points A, B, and C. e. Write the equation of the plane containing the points A, B, and C. f. Write the equation of the line passing through points D and E. g. Find the point where the line through D and E intersects the plane through points A, B, and C. h. Find the distance from the point D to the plane through points A, B, and C. i. Write the equation of the sphere with center D that is tangent to the plane through points A, B, and C. 2. The region R is bounded by the lines x = 4 , y = 7 , and y = 2x + 5 . a. Find the volume of the solid formed when R is rotated about the x-axis. b. Find the volume of the solid formed when R is rotated about the y-axis. c. Find the volume of the solid formed when R is rotated about the line y = 14 . d. Find the volume of the solid formed when R is rotated about the line x = −2 .

MA Notes 18 Problem Sets

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Math Analysis Problem Set 18-02 This problem is probably a ton of work…If you can do it then you can stop worrying about studying for these sorts of things.

1. Write equations that define each edge of the region. 2. We’re going to revolve the region around the y-axis. Break the region into five horizontal strips in such a way that the lines forming the left and right edges of the strips are consistent as you move vertically through the strip. For instance, the first strip would be from y = 3 to y = 4 . 3. Find the volume of the solid formed as each strip is rotated about the y-axis. For each strip, make sure to write a plan (Cone – cone – cylinder or something like it) and then list the relevant dimensions of each element of your plan. Finally carry out the plan. 4. Sum up the volumes you found in the previous problem to find the volume of the entire solid. The total volume might be 246π or I might have made a mistake…

MA Notes 18 Problem Sets

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Math Analysis Problem Set 18-03 The numerical approximations that we’ve been doing are actually called Riemann Sums. They’re a big, big deal in calculus (though in a slightly modified version from the way we’re using them). They can be used to approximate all sorts of things. x + 3 , and x = 4 . 3 1. Sketch the region and find all necessary intersection points.

The region R is bounded by the curves y = x + 2 , y =

2. The area problem. a. Write a summation that approximates the area of R using n rectangles. b. Approximate the area of R using 10, 100, and 1000 rectangles. 3. Rotate around the x-axis. a. Write a summation that approximates the volume of the solid formed when the region R is rotated about the x-axis. b. Approximate the volume of the solid using 10, 100, and 1000 cylinders. 4. Rotate around y = −3 . a. Write a summation that approximates the volume of the solid formed when the region R is rotated about the line y = −3 . b. Approximate the volume of the solid using 10, 100, and 1000 cylinders. 5. Around the y-axis. a. Write a summation that approximates the volume of the solid formed when the region R is rotated about the y-axis. This is more complicated because the region is not horizontally simple—if you take a horizontal line and pass it through the region the line doesn’t constantly intersect the same curve on the right side or the same curve on the left side. (The region is vertically simple; imagine a vertical line passing through the region.) b. Approximate the volume using 10, 100, and 1000 cylinders. 6. R is the base of a solid whose cross-sections perpendicular to the x-axis are squares. a. Write a summation that approximates the volume of the solid using n square prisms. b. Approximate the volume of the solid using 10, 100, and 1000 prisms. 7. R is the base of a solid whose cross-sections perpendicular to the x-axis are equilateral triangles. a. Write a summation that approximates the volume of the solid using n triangular prisms. b. Approximate the volume of the solid using 10, 100, and 1000 prisms.

MA Notes 18 Problem Sets

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Here’s a picture of what the solid looks like when the cross-sections are squares.

Here’s a picture of what the solid looks like when the cross-sections are equilateral triangles.

MA Notes 18 Problem Sets

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